Iterative decoding

ABSTRACT

This invention provides an iterative process to maximum a posteriori (MAP) decoding. The iterative process uses an auxiliary function which is defined in terms of a complete data probability distribution. The auxiliary function is derived based on an expectation maximization (EM) algorithm. For a special case of trellis coded modulators, the auxiliary function may be iteratively evaluated by a combination of forward-backward and Viterbi algorithms. The iterative process converges monotonically and thus improves the performance of any decoding algorithm. The MAP decoding minimizes a probability of error. A direct approach to achieve this minimization results in complexity which grows exponentially with T, where T is the size of the input. The iterative process avoids this complexity by converging on the MAP solution through repeated maximization of the auxiliary function.

This nonprovisional application claims the benefit of U.S. provisional application No. 60/174,601 entitled “Map Decoding In Channels With Memory” filed on Jan. 5, 2000. The Applicant of the provisional application is William Turin. The above provisional application is hereby incorporated by reference including all references cited therein.

BACKGROUND OF THE INVENTION

1. Field of Invention

This invention relates to iterative decoding of input sequences.

2. Description of Related Art

Maximum a posteriori (MAP) sequence decoding selects a most probable information sequence X₁ ^(T)=(X₁, X₂, . . . , X_(T)) that produced the received sequence Y₁ ^(T)=(Y₁, Y₂, . . . , Y_(T)). For transmitters and/or channels that are modeled using Hidden Markov Models (HMM), the process for obtaining the information sequence X₁ ^(T) that corresponds to a maximum probability is difficult due to a large number of possible hidden states as well as a large number of possible information sequences X₁ ^(T). Thus, new technology is needed to improve MAP decoding for HMMs.

SUMMARY OF THE INVENTION

This invention provides an iterative process to maximum a posteriori (MAP) decoding. The iterative process uses an auxiliary function which is defined in terms of a complete data probability distribution. The MAP decoding is based on an expectation maximization (EM) algorithm which finds the maximum by iteratively maximizing the auxiliary function. For a special case of trellis coded modulation, the auxiliary function may be maximized by a combination of forward-backward and Viterbi algorithms. The iterative process converges monotonically and thus improves the performance of any decoding algorithm.

The MAP decoding decodes received inputs by minimizing a probability of error. A direct approach to achieve this minimization results in a complexity which grows exponentially with T, where T is the size of the input. The iterative process avoids this complexity by converging on the MAP solution through repeated use of the auxiliary function.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is described in detail with reference to the following figures where like numerals reference like elements, and wherein:

FIG. 1 shows a diagram of a communication system;

FIG. 2 shows a flow chart of an exemplary iterative process;

FIGS. 3-6 show state trajectories determined by the iterative process;

FIG. 7 shows an exemplary block diagram of the receiver shown in FIG. 1;

FIG. 8 shows a flowchart for an exemplary process of the iterative process for a TCM example; and

FIG. 9 shows step 1004 of the flowchart of FIG. 8 in greater detail.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

FIG. 1 shows an exemplary block diagram of a communication system 100. The communication system 100 includes a transmitter 102, a channel 106 and a receiver 104. The transmitter 102 receives an input information sequence I₁ ^(T) (i.e., I₁, I₂, . . . , I_(T)) of length T, for example. The input information sequence may represent any type of data including analog voice, analog video, digital image, etc. The transmitter 102 may represent a speech synthesizer, a signal modulator, etc.; the receiver 104 may represent a speech recognizer, a radio receiver, etc.; and the channel 106 may be any medium through which the information sequence X₁ ^(T) (i.e., X₁, X₂, . . . , X_(T)) is conveyed to the receiver 104. The transmitter 102 may encode the information sequence I₁ ^(T) and transmit encoded information sequence X₁ ^(T) through the channel 106 and the receiver 104 receives information sequence Y₁ ^(T) (i.e., Y₁, Y₂, . . . , Y_(T)). The problem in communications is, of course, to decode Y₁ ^(T) in such a way as to retrieve I₁ ^(T).

Maximum a posteriori (MAP) sequence decoding is a technique that decodes the received sequence Y₁ ^(T) by minimizing a probability of error to obtain X₁ ^(T) (and if a model of the transmitter 102 is included, to obtain I₁ ^(T)). In MAP, the goal is to choose a most probable X₁ ^(T) that produces the received Y₁ ^(T). The MAP estimator may be expressed by equation 1 below. $\begin{matrix} {{\hat{X}}_{1}^{T} = {\underset{X_{1}^{T}}{\arg \quad \max}\quad \Pr \quad \left( {X_{1}^{T},Y_{1}^{T}} \right)}} & (1) \end{matrix}$

where Pr(·) denotes a corresponding probability or probability density function and {circumflex over (X)}₁ ^(T) is an estimate of X₁ ^(T). Equation 1 sets {circumflex over (X)}₁ ^(T) to the X₁ ^(T) that maximizes Pr(X₁ ^(T), Y₁ ^(T)).

The Pr(X₁ ^(T), Y₁ ^(T)) term may be obtained by modeling the channel 106 of the communication system 100 using techniques such as Hidden Markov Models (HMMs). An input-output HMM λ=(S,X,Y,π, {P(X,Y)}) is defined by its internal states S={1,2, . . . n}, inputs X, outputs Y, initial state probability vector π, and the input-output probability density matrices (PDMs) P(X,Y), XεX, YεY. The elements of P(X,Y), p_(ij)(X,Y)=Pr(j,X,Y|i), are conditional probability density functions (PDFs) of input x and corresponding output y after transferring from the state i to state j. It is assumed that the state sequence S₀ ^(t)=(S₀, S₁, . . . , S_(t)), input sequence X₁ ^(t)=(X₁,X₂, . . . X_(t)), and output sequence Y₁ ^(t)=(Y₁,Y₂, . . . , Y_(t)) possess the following Markovian property

Pr(S _(t) ,X _(t) ,Y _(t) |S ₀ ^(t−1) ,X ₁ ^(t−1) ,Y ₁ ^(t−1))=Pr(S _(t) ,X _(t) ,Y _(t) |S _(t−1)).

Using HMM, the PDF of the input sequence X₁ ^(T) and output sequence Y₁ ^(T) may be expressed by equation 2 below: $\begin{matrix} {{p_{T}\quad \left( {X_{1}^{T},Y_{1}^{T}} \right)} = {\pi \quad {\prod\limits_{i = 1}^{T}\quad {P\quad \left( {X_{i},Y_{i}} \right)\quad 1}}}} & (2) \end{matrix}$

where 1 is a column vector of n ones, π is a vector of state initial probabilities, and n is a number of states in the HMM. Thus, the MAP estimator when using HMM may be expressed by equation 3 below: $\begin{matrix} {{\hat{X}}_{1}^{T} = {\arg \quad {\max\limits_{X_{1}^{T}}\left\lbrack {\pi \quad {\prod\limits_{i = 1}^{T}\quad {P\quad \left( {X_{i},Y_{i}} \right)\quad 1}}} \right\rbrack}}} & (3) \end{matrix}$

The maximization required by equation 3 is a difficult problem because all possible sequences of X₁ ^(T) must be considered. This requirement results in a complexity that grows exponentially with T. This invention provides an iterative process to obtain the maximum without the complexity of directly achieving the maximization by evaluating equation 2 for all possible X₁ ^(T), for example. In the iterative process, an auxiliary function is developed whose iterative maximization generates a sequence of estimates for X₁ ^(T) approaching the maximum point of equation 2.

The iterative process is derived based on the expectation maximization (EM) algorithm. Because the EM algorithm converges monotonically, the iterative process may improve the performance of any decoding algorithm by using its output as an initial sequence of the iterative decoding algorithm. In the following description, it is assumed that HMM parameters for the channel 106 and/or the transmitter 102 are available either by design or by techniques such as training.

The auxiliary function may be defined in terms of a complete data probability distribution shown in equation 4 below. $\begin{matrix} {{{\Psi \quad \left( {z,X_{1}^{T},Y_{1}^{T}} \right)} = {\pi_{i_{o}}\quad {\prod\limits_{i = 1}^{T}\quad {p_{i_{t - 1}i_{t}}\quad \left( {X_{t},Y_{t}} \right)}}}},} & (4) \end{matrix}$

where z=i₀ ^(T) is an HMM state sequence, π_(i) ₀ is an initial probability vector for state i₀, and p_(ij)(X,Y) are the elements of the matrix P(X,Y). The MAP estimator of equation 1 can be obtained iteratively by equations 5-9 as shown below. $\begin{matrix} {{X_{1,{p + 1}}^{T} = {\arg \quad {\max\limits_{X_{1}^{T}}\quad {Q\quad \left( {X_{1}^{T},X_{1,p}^{T}} \right)}}}},{p = 0},1,2,\ldots} & (5) \end{matrix}$

where p is a number of iterations and Q(X_(i) ^(t),X_(i,p) ^(t)) is the auxiliary function which may be expressed as $\begin{matrix} {{Q\quad \left( {X_{1}^{T},X_{1,p}^{T}} \right)} = {\sum\limits_{z}\quad {\Psi \quad \left( {z,X_{1,p}^{T},Y_{1}^{T}} \right)\quad \log \quad {\left( {\Psi \quad \left( {z,X_{1}^{T},Y_{1}^{T}} \right)} \right).}}}} & (6) \end{matrix}$

The auxiliary function may be expanded based on equation 4 as follows: $\begin{matrix} {{Q\quad \left( {X_{1}^{T},X_{1,p}^{T}} \right)} = {{\sum\limits_{t = 1}^{T}\quad {\sum\limits_{i = 1}^{n}\quad {\sum\limits_{j = 1}^{n}\quad {\gamma_{t,{ij}}\quad \left( X_{1,p}^{T} \right)\quad \log \quad \left( {p_{ij}\quad \left( {X_{t},Y_{t}} \right)} \right)}}}} + C}} & (7) \end{matrix}$

where C does not depend on X₁ ^(T), n is a number of states in the HMM and

γ _(t,ij)(X _(1,p) ^(T))=α_(i)(X _(1,p) ^(t−1) ,Y ₁ ^(t−1))P _(ij)(X _(t,p) ,Y _(t))β_(j)(X _(t+1,p) ^(T) ,Y _(t+1) ^(T))  (8)

where α_(i)(X_(1,p) ^(t),Y₁ ^(t)) and β_(j)(X_(t+1,p) ^(T),Y_(t+1) ^(T)) are the elements of the following forward and backward probability vectors $\begin{matrix} {{{\alpha \quad \left( {X_{1}^{T},Y_{1}^{T}} \right)} = {\pi \quad {\prod\limits_{i = 1}^{T}\quad {P\quad \left( {X_{i},Y_{i}} \right)}}}},{{{and}\quad \beta \quad \left( {X_{t + 1}^{T},Y_{t + 1}^{T}} \right)} = {\prod\limits_{t + 1}^{T}\quad {P\quad \left( {X_{i},Y_{i}} \right)\quad 1.}}}} & (9) \end{matrix}$

Based on equations 5-9, the iterative process may proceed as follows. At p=0, an initial estimate of X_(1,0) ^(T) is generated. Then, Q(X₁ ^(T),X_(1,0) ^(T)) is generated for all possible sequences of X₁ ^(T). From equations 7 and 8, Q(X₁ ^(T),X_(1,0) ^(T)) may be evaluated by generating γ_(t,ij)(X_(1,0) ^(T)) and log(p_(ij)(X_(t),Y^(t))) for each t, i, and j. γ_(t,ij)(X_(1,0) ^(T)) may be generated by using the forward-backward algorithm as shown below:

α(X _(1,p) ⁰ ,Y ₁ ⁰)=π, α(X _(t,p) ^(t) ,Y ₁ ^(t))=α(X _(1,p) ^(t−1) ,Y ₁ ^(t−1))P(X _(t,p) ,Y _(t)),t=1,2, . . . T

β(X _(T+1,p) ^(T) ,Y _(T+1) ^(T))=1, β(X _(t,p) ^(T) ,Y _(t) ^(T))=P(X _(t,p) ,Y _(t))β(X _(t+1,p) ^(T) ,Y _(t+1) ^(T)), t=T−1,T−2, . . . , 1

Log(p^(ij)(X_(t),Y_(t)))is generated for all possible X_(t) for t=1,2, . . . , T and the (X_(t))s that maximize D(X₁ ^(T), X_(1,0) ^(T)) are selected as X_(1,1) ^(T). After X_(1,1) ^(T) is obtained, it is compared with X_(1,0) ^(T). If a measure D(X_(1,1) ^(T),X_(1,0) ^(T)) of difference between the sequences exceeds a compare threshold, then the above process is repeated until the difference measure D(X_(1,p) ^(T),X_(1,p−1) ^(T)) is within the threshold. The last X_(1,p) ^(T) for p iterations is the decoded output. The measure of difference may be an amount of mismatch information. For example, if X₁ ^(T) is a sequence of symbols, then the measure may be a number of different symbols between X_(1,p) ^(T) and X_(1,p−1) ^(T) (Hamming distance); if X₁ ^(T) is a sequence of real numbers, then the measure may be an Euclidean distance D(X_(1,p) ^(T),X_(1,p−1) ^(T))=[Σ_(i=1) ^(T)(X_(i,p)−X_(i,p−1))²]^(½).

FIG. 2 shows a flowchart of the above-described process. In step 1000, the receiver 104 receives the input information sequence Y₁ ^(T) and goes to step, 1002. In step 1002, the receiver 104 selects an initial estimate for the decode output information sequence X_(1,0) ^(T) and goes to step 1004. In step 1004, the receiver 104 generates γ_(t,ij)(X_(1,p) ^(T)) where p=0 for the first iteration and goes to step 1006. In step 1006, the receiver 104 generates all the log(p_(ij)(X_(t),Y_(t))) values and goes to step 1008.

In step 1008, the receiver 104 selects a sequence X_(1,p+1) ^(T) that maximizes Q(X_(1,p+1) ^(T),X_(1,p) ^(T)) and goes to step 1010. In step 1010, the receiver 104 compares X_(1,p) ^(T) with X_(1,p+1) ^(T). If the compare result is within the compare threshold, then the receiver 104 goes to step 1012; otherwise, the receiver 104 returns to step 1004 and continues the process with the new sequence X_(1,p+1) ^(T). In step 1012, the receiver 104 outputs X_(1,p+1) ^(T) and goes to step 1014 and ends the process.

The efficiency of the above described iterative technique may be improved if the transmitted sequence is generated by modulators such as a trellis coded modulator (TCM). A TCM may be described as a finite state machine that may be defined by equations 10 and 11 shown below.

S _(t+1) =f _(t)(S _(t) ,I _(t))  (10)

X _(t) =g _(t)(S _(t) ,I _(t))  (11)

Equation 10 specifies the TCM state transitions while equation 11 specifies the transmitted information sequence based on the state and the input information sequence. For example, after receiving input I_(t) in state S_(t), the finite state machine transfers to state S_(t+1) based on S_(t) and I_(t) as shown in equation 10. The actual output by the transmitter 102 is X_(t) according to equation 11. Equation 10 may represent a convolutional encoder and equation 11 may represent a modulator. For the above example, the transmitter output information sequence may not be independent even if the input information sequence X₁ ^(T) is independent.

In equation 15, the log(p_(ij)(Y_(t),X_(t))) term may be analyzed based on the TCM state transitions because the information actually transmitted X_(t) is related to the source information I_(t) by X_(t)=g_(t)(S_(t),I_(t)). This relationship between X_(t) and I_(t) forces many elements p_(ij)(Y_(t),X_(t)) of P(Y_(t),X_(t)), to zero since the finite state machine (equations 10 and 11) removes many possibilities that otherwise must be considered. Thus, unlike the general case discussed in relation to equations 5-9, evaluation of p_(ij)(Y_(t),X_(t)) may be divided into a portion that is channel related and another portion that is TCM related. The following discussion describes the iterative technique in detail for the TCM example.

For a TCM system with an independent and identically distributed information sequence, an input-output HMM may be described by equations 12 and 13 below.

P(X _(t) ,Y _(t))=[p _(s) _(t) _(s) _(t+1) P _(c)(X _(t) |Y _(t))],  (12)

where $\begin{matrix} {p_{s_{t}s_{t + 1}} = \left\{ \begin{matrix} {\Pr \quad \left( I_{t} \right)} & {{{if}\quad S_{t + 1}} = {f_{t}\quad \left( {S_{t},I_{t}} \right)}} \\ 0 & {otherwise} \end{matrix} \right.} & (13) \end{matrix}$

P_(c)(Y₁|X_(t)) is the conditional PDM of receiving Y_(t) given that X_(t) has been transmitted for the HMM of a medium (channel) through which the information sequence is transmitted; p_(s) _(t) _(s) _(t+1) is the probability of the TCM transition from state S_(t) to state S_(t+1), and Pr(I_(t)) is the probability of an input I_(t). Thus, equation 2 may be written as ${{p_{T}\quad \left( {I_{1}^{T},Y_{1}^{T}} \right)} = {\pi_{c}\quad {\prod\limits_{i = 1}^{T}\quad {p_{s_{t}s_{t + 1}}\quad P_{c}\quad \left( Y_{t} \middle| X_{t} \right)\quad 1}}}},$

where π_(c) is a vector of the initial probabilities of the channel states, X_(t)=g_(t)(S_(t),I_(t)), and the product is taken along the state trajectory S_(t+1)=f_(t)(S_(t),I_(t)) for t=1,2, . . . , T.

If all elements of the input information sequence are equally probable, then the MAP estimate may be expressed by equation 14 below. $\begin{matrix} {{{\hat{I}}_{1}^{T} = {\arg \quad {\max\limits_{I_{1}^{T}}\quad {\pi_{c}\quad {\prod\limits_{t = 1}^{T}\quad {P_{c}\quad \left( Y_{t} \middle| X_{t} \right)\quad 1}}}}}},} & (14) \end{matrix}$

The auxiliary function may be expressed by equations 15-17 below corresponding to equations 7-9 above. $\begin{matrix} {{Q\quad \left( {I_{1}^{T},I_{1,p}^{T}} \right)} = {{\sum\limits_{t = 1}^{T}\quad {\sum\limits_{i = 1}^{n}\quad {\sum\limits_{j = 1}^{n}\quad {\gamma_{t,{ij}}\quad \left( I_{1,p}^{T} \right)\quad \log \quad \left( {p_{ij}\quad \left( Y_{t} \middle| X_{t} \right)} \right)}}}} + C}} & (15) \end{matrix}$

where X_(t)=g_(t)(S_(t),I_(t)) and

γ_(t,ij)(I _(1,p) ^(T))=α_(i)(Y ₁ ^(t−1) |I _(1,p) ^(t))P _(c,ij)(Y _(t) |X _(t,p))β_(j)(Y _(t+1,p) ^(T) |I _(t+1,p) ^(T))  (16)

α_(i)(Y₁ ^(T−1)|I_(1,p) ^(t−1)) and β_(j)(Y_(t+1,p) ^(T)|I_(t+1,p) ^(T))

are the elements of the forward and backward probability vectors $\begin{matrix} {{\alpha \quad \left( Y_{1}^{t} \middle| I_{1}^{T} \right)} = {{\pi_{c}\quad {\prod\limits_{i = 1}^{t}\quad {P_{c}\quad \left( Y_{i} \middle| X_{i} \right)\quad {and}\quad \beta \quad \left( Y_{t + 1}^{T} \middle| I_{t + 1}^{T} \right)}}} = {\prod\limits_{i = {t + 1}}^{T}\quad {P_{c}\quad \left( Y_{i} \middle| X_{i} \right)\quad 1}}}} & (17) \end{matrix}$

From equation 15, the Viterbi algorithm may be applied with the branch metric $\begin{matrix} {{{m\quad \left( I_{t} \right)} = {\sum\limits_{i = 1}^{n}\quad {\sum\limits_{j = 1}^{n}\quad {\gamma_{t,{ij}}\quad \left( I_{1,p}^{T} \right)\quad \log \quad p_{c,{ij}}\quad \left( Y_{t} \middle| X_{t} \right)}}}},\quad {t = 1},2,\ldots \quad,T} & (18) \end{matrix}$

to find a maximum of Q(I₁ ^(T),I_(1,p) ^(T)) which can be interpreted as a longest path leading from the initial zero state to one of the states S_(T) where only the encoder trellis is considered. The Viterbi algorithm may be combined with the backward portion of the forward-backward algorithm as follows.

1. Select an initial source information sequence I_(1,0) ^(T)=I_(1,0),I_(2,0), . . . , I_(T,0)

2. Forward part:

a. set α(Y₁ ⁰|I₁ ⁰)=π, where π is an initial state probability estimate; and

b. for t=1,2, . . . ,T, compute X_(t,p)=g_(t)(S_(t),I_(t,p)), α(Y₁ ^(t)|I_(1,p) ^(t))=α(Y_(1,p) ^(t−1)|I_(1,p) ^(t−1))P_(c)(Y_(t)|X_(1,p)), where I_(1,p) ^(t) is a prior estimate of I₁ ^(t).

3. Backward part:

a. setβ(Y_(T+1) ^(T)|I_(T+1,p) ^(T))=1 and last state transition lengths L(S_(T)) to 0 for all the states;

 for t=T,T−1, . . . , 1 compute:

b. X_(t)=g_(t)(S_(t),I_(t)),

c. γ_(t,ij)(I_(1,p) ^(T))=α_(i)(Y₁ ^(t−1)|I^(t−1))P_(c,j) (Y_(t)|X_(t,p))β_(j)(Y_(t+1) ^(T)|I_(t+1,p) ^(T)),

d. ${L\left( S_{t} \right)} = {\max\limits_{I_{t}}{\left\{ {{L\left\lbrack {f_{t}\left( {S_{t},I_{t}} \right)} \right\rbrack} + {m\left( I_{t} \right)}} \right\}.}}$

This step selects the paths with the largest lengths (the survivors).

e. ${{\hat{I}}_{t}\left( S_{t} \right)} = {\arg \quad {\max\limits_{I_{t}}{\left\{ {{L\left\lbrack {f_{t}\left( {S_{t},I_{t}} \right)} \right\rbrack} + {m\left( I_{t} \right)}} \right\}.}}}$

This step estimates I_(t) corresponding to the state S_(t) by selecting the I_(t) of the survivor in step d.

f. β(Y_(t) ^(T)|I_(t,p) ^(T))=P_(c)(Y_(t)|X_(t,p)) β(Y_(t+1) ^(T)|X_(t+1,p) ^(T)).

g. End (of “for” loop).

4. Reestimate the information sequence:

I_(t,p+1)=Î_(t)(Ŝ_(t)), Ŝ_(t+1)=f_(t)(Ŝ_(t),I_(t,p+1)), t=1,2, . . . , T where Ŝ₁=0; and

5. If I_(t,p+1)≠I_(t,p), go to step 2; otherwise decode the information sequence as I_(t,p+1) ^(T).

FIGS. 3-6 show an example of the iterative process discussed above where there are four states in the TCM and T=5. The dots represent possible states and the arrows represent a state trajectory that corresponds to a particular information sequence. The iterative process may proceed as follows. First, an initial input information sequence I_(1,0) ⁵ is obtained. I_(1,0) ⁵ may be the output of an existing decoder or may simply be a guess.

The Viterbi algorithm together with the backward algorithm may be used to obtain a next estimate of the input information sequence I_(1,1) ⁵. This process begins with the state transitions between t=4 and t=5 by selecting state transitions leading to each of the states s0-s3 at t=4 from states at t=5 that have the largest value of the branch metric L(S₄)=m(I₅) of equation 18 above. Then, the process moves to select state transitions between the states at t=3 and t=4 that have the largest cumulative distance L(S₃)=L(S₄)+m(I₄). This process continues until t=0 and the sequence of input information I₁ ⁵ corresponding to the path connecting the states from t=0 to t=5 that has the longest path ${{L\quad \left( S_{0} \right)} = \sum\limits_{t = 1}^{5}}\quad$

m(I_(t)) is selected as the next input information sequence I_(1,1) ⁵.

For the example in FIG. 3, state transitions from the states at t=4 to all the states at t=5 are considered. Assuming that the (I_(t))s are binary, then only two transitions can emanate from each of the states at t=4: one transition for I₅=0 and one transition for I₅=1. Thus, FIG. 3 shows two arrows terminating on each state at t=4 (arrows are “backwards” because the backward algorithm is used). State transitions 301 and 302 terminate at state s0; state transitions 303 and 304 terminate at state s1; state transitions 305 and 306 terminate at state s2; and state transitions 307 and 308 terminate at state s3.

The branch metric m(I_(t)) of equation 18 represents a “distance” between the states and is used to select the state transition that corresponds to the longest path for each of the states s0-s3 at t=4: $\begin{matrix} \begin{matrix} {{m\quad \left( I_{5} \right)} = \quad {\sum\limits_{i}^{\quad}\quad {\sum\limits_{j}^{\quad}\quad {\gamma_{5,{ij}}\quad \left( X_{1,0}^{5} \right)\quad \log \quad p_{ij}\quad \left( {X_{5},Y_{5}} \right)}}}} \\ {{= \quad {\sum\limits_{i}^{\quad}\quad {\sum\limits_{j}^{\quad}\quad {\alpha_{i}\quad \left( {X_{1,0}^{4},Y_{1}^{4}} \right)\quad p_{ij}\quad \left( {X_{5,0},Y_{5}} \right)\quad \beta_{j}\quad \left( {X_{5,0},Y_{5}} \right)\quad \beta_{j}\quad \left( {X_{6,0}^{5},Y_{6}^{5}} \right)\quad \log \quad p_{ij}\quad \left( Y_{5} \middle| X_{5} \right)}}}},} \end{matrix} & (19) \end{matrix}$

where β_(j)(X_(6,0) ⁵,Y₆ ⁵)=1, and X₅=g₅(S₅,I₅) by definition. There is an I₅ that corresponds to each of the state transitions 301-308. For this example, L(S₄)=m(I₅) corresponding to odd numbered state transitions 301-307 are greater than that for even numbered state transitions 302-308. Thus, odd numbered state transitions are “survivors.” Each of them may be part of the state trajectory that has the longest path from t=0 to t=5. This transition (the survivor) is depicted by the solid arrow while the transitions with smaller lengths are depicted by dashed lines.

The state sequence determination process continues by extending the survivors to t=3 as shown in FIG. 4 forming state transitions 309-316. The distance between state transitions for each of the states are compared based on L(S₄)+m(I₄), where m(I₄) is shown in equation 20 below. $\begin{matrix} \begin{matrix} {{m\quad \left( I_{4} \right)} = \quad {\sum\limits_{i}^{\quad}\quad {\sum\limits_{j}^{\quad}\quad {\gamma_{5,{ij}}\quad \left( X_{1,0}^{5} \right)\quad \log \quad p_{ij}\quad \left( {X_{4},Y_{4}} \right)}}}} \\ {= \quad {\sum\limits_{i}^{\quad}\quad {\sum\limits_{j}^{\quad}\quad {\alpha_{i}\quad \left( {X_{1,0}^{3},Y_{1}^{3}} \right)\quad p_{ij}\quad \left( {X_{4,0},Y_{4}} \right)\quad \beta_{j}\quad \left( {X_{5,0},Y_{5}} \right)\quad \log \quad p_{ij}\quad {\left( Y_{4} \middle| X_{4} \right).}}}}} \end{matrix} & (20) \end{matrix}$

For this example, the distances corresponding to the odd numbered state transitions 309-315 are longer than distances corresponding to even numbered state transitions 310-316. Thus, the paths corresponding to the odd numbered state transitions are the survivors. As shown in FIG. 4, the state transition 301 is not connected to any of the states at t=3 and thus is eliminated even though it was a survivor. The other surviving state transitions may be connected into partial state trajectories. For example, partial state trajectories are formed by odd numbered state transitions 307-309, 303-311, 303-313 and 305-315.

The above process continues until t=0 is reached as shown in FIG. 5 where two surviving state trajectories 320-322 are formed by the surviving state trajectories. All the state trajectories terminate at state zero for this example because, usually, encoders start at state zero. As shown in FIG. 6, the state trajectory that corresponds to the longest cumulative distance is selected and the input information sequence I₁ ⁵(via S_(t+1)=f_(t)(S_(t)I_(t)) that corresponds to the selected trajectory is selected as the next estimated input information sequence Î_(1,1) ⁵. For this example, the state trajectory 320 is selected and the input information sequence I₁ ⁵ corresponding to the state trajectory 320 is selected as Î_(1,1) ⁵.

FIG. 7 shows an exemplary block diagram of the receiver 104. The receiver 104 may include a controller 202, a memory 204, a forward processor 206, a backward processor 208, a maximal length processor 210 and an input/output device 212. The above components may be coupled together via a signal bus 214. While the receiver 104 is illustrated using a bus architecture, any architecture may be suitable as is well known to one of ordinary skill in the art.

All the functions of the forward, backward and maximal length processors 206, 208 and 210 may also be performed by the controller 202 which may be either a general purpose or special purpose computer (e.g., DSP). FIG. 7 shows separate processors for illustration only. The forward, backward maximal length processors 206, 208 and 210 may be combined and may be implemented by using ASICs, PLAs, PLDs, etc. as is well known in the art.

The forward processor 206 generates the forward probability vectors α_(i)(X_(1,p) ^(t−1),Y₁ ^(t−1)) herein referred to as α_(i). For every iteration, when a new X_(1,p) ^(T)(or I_(1,p) ^(T)) is generated, the forward processor 206 may generate a complete set of α_(i).

The backward processor 208 together with the maximal length processor 210 generate a new state sequence by searching for maximal length state transitions based on the branch metric m(I_(t)). Starting with the final state transition between states corresponding to t=T−1 and t=T, the backward processor generates β_(j)(X_(t+1,p) ^(T),Y_(t+1) ^(T)) (hereinafter referred as β_(j)) as shown in equation 8 for each state transition.

The maximal length processor 210 generates m(I_(t)) based on the results of the forward processor 206, the backward processor 208 and p_(ij)(X₁,Y_(t)). After generating all the m(I_(t))s corresponding to each of the possible state transitions, the maximal length processor 210 compares all the L(S_(t))+m(I_(t))s and selects the state transition that corresponds to the largest L(S_(t))+m(I_(t)), and the I_(t)(via S_(t+1)=f_(t)(S_(t),I_(t))) that corresponds to the selected state transition is selected as the estimated input information for that t. The above process is performed for each t=1,2, . . . , T to generate a new estimate I_(1,p) ^(T) for each of the iteration p.

Initially, the controller 202 places an estimate of the PDM P(X,Y) and π in the memory 204 that corresponds to the HMM for the channel 106 and/or the transmitter 102. The PDM P(X,Y) may be obtained via well known training processes, for example.

When ready, the controller 202 receives the received input information sequence Y₁ ^(T) and places them in the memory 204 and selects an initial estimate of I_(1,0) ^(T)(or X_(1,0) ^(T)). The controller 202 coordinates the above-described iterative process until a new estimate I_(1,1) ^(T) (or X_(1,1) ^(T)) is obtained. Then, the controller 202 compares I_(1,0) ^(T) with I_(1,1) ^(T) to determine if the compare result is below the compare threshold value (e.g., matching a predetermined number of elements or symbols of the information sequence). The compare threshold may be set to 0, in which case I_(1,0) ^(T) must be identical with I_(1,1) ^(T). If an acceptable compare result is reached, I_(1,1) ^(T) is output as the decoded output. Otherwise, the controller 202 iterates the above-described process again and compares the estimated I_(1,p) ^(T) with I_(1,p−1) ^(T) until an acceptable result is reached and I_(1,p) ^(T) is output as the decoded output.

FIG. 8 shows a flowchart of the above-described process. In step 1000, the controller 202 receives Y₁ ^(T) via the input/output device 212 and places Y₁ ^(T) in the memory 204 and goes to step 1002. In step 1002, the controller 202 selects an initial estimate for I_(1,0) ^(T) and goes to step 1004. In step 1004, the controller 202 determines a new state sequence and a next estimated I_(1,1) ^(T) (I_(1,p) ^(T), where p=1) (via the forward, backward and maximal length processors 206, 208 and 210) and goes to step 1006. In step 1006, the controller 202 compares I_(1,0) ^(T) with I_(1,1) ^(T). If the compare result is within the predetermined threshold, then the controller 202 goes to step 1008; otherwise, the controller 202 returns to step 1004. In step 1008, the controller 202 outputs I_(1,p) ^(T) where p is the index of the last iteration and goes to step 1010 and ends the process.

FIG. 9 shows a flowchart that expands step 1004 in greater detail. In step 2000, the controller 202 instructs the forward processor 206 to generate α_(i) as shown in equation 8, and goes to step 2002. In step 2002, the controller 202 sets the parameter t=T and goes to step 2004. In step 2004, the controller 202 instructs the backward processor 208 to generate β_(j) and the maximal length processor 210 to determine next set of survivors based on equation 18 and time t+1 survivors and goes to step 2006.

In step 2006, the controller 202 decrements t and goes to step 2008. In step 2008, the controller 202 determines whether t is equal to 0. If t is equal to 0, the controller 202 goes to step 2010; otherwise, the controller 202 returns to step 2004. In step 2010, the controller 202 outputs the new estimated I₁ ^(T) and goes to step 2012 and returns to step 1006 of FIG. 5.

A specific example of the iterative process for convolutional encoders is enclosed in the appendix.

While this invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications, and variations will be apparent to those skilled in the art. Accordingly, preferred embodiments of the invention as set forth herein are intended to be illustrative, not limiting. Various changes may be made without departing from the spirit and scope of the invention.

For example, a channel may be modeled as P_(c)(Y|X)=P_(c)B_(c)(Y|X) where P_(c) is a channel state transition probability matrix and B_(c)(Y|X) is a diagonal matrix of state output probabilities. For example, based on the Gilbert-Elliott model ${{B_{c}\quad \left( X \middle| X \right)} = {{\begin{bmatrix} {1 - b_{1}} & 0 \\ 0 & {1 - b_{2}} \end{bmatrix}\quad {and}\quad B_{c}\quad \left( \overset{\_}{X} \middle| X \right)} = \begin{bmatrix} b_{1} & 0 \\ 0 & b_{2} \end{bmatrix}}},$

where {overscore (X)} is the complement of X. For this case, m(I_(t)) may be simplified as ${{m\quad \left( I_{t} \right)} = {\sum\limits_{i = 1}^{n_{c}}\quad {\gamma_{t,i}\quad \left( I_{1,p}^{T} \right)\quad b_{j}\quad \left( Y_{t} \middle| X_{t} \right)}}},\quad {t = 1},2,\ldots \quad,T,\quad {and}$

 γ_(t,i)(I _(1,p) ^(T))=α_(i)(Y ₁ ^(t) |I _(1,p) ^(t))β_(i)(Y _(t+1) ^(T) |I _(t+1,p) ^(T)), where b _(j)(Y _(t) |X _(t)) are the elements of B_(c). 

What is claimed is:
 1. A method for maximum a posteriori (MAP) decoding of an input information sequence based on a first information sequence received through a channel, comprising: iteratively generating decode results, X_(i), for i=1,2, . . . n, where n is an integer, and where each X_(i) is generated by employing X_(i−1) and X₁ is generated from said first information sequence and from initial decode result, X₀; and ceasing said step of iteratively generating, and outputting last-generated decode results when difference between said last-generated decode results and next-to-last-generated decode results is within a compare threshold.
 2. A method for maximum a posteriori (MAP) decoding of an input information sequence based on a first information sequence received through a channel, comprising: iteratively generating a sequence of one or more decode results starting with an initial decode result; and outputting one of adjacent decode results as a decode of the input information sequence if the adjacent decode results are within a compare threshold, wherein the step of iteratively generating comprises: a. generating the initial decode result as a first decode result; b. generating a second decode result based on the first decode result and a model of the channel; c. comparing the first and second decode results; d. replacing the first decode result with the second decode result; and e. repeating b-d if the first and second decode results are not within the compare threshold.
 3. The method of claim 2, wherein the generating a second decode result comprises searching for a second information sequence that maximizes a value of an auxiliary function.
 4. The method of claim 3, wherein the auxiliary function is based on the expectation maximization (EM) algorithm.
 5. The method of claim 4, wherein the model of the channel is a Hidden Markov Model (HMM) having an initial state probability vector π and probability density matrix (PDM) of P(X,Y), where XεX, YεY and elements of P(X,Y), p_(ij)(X,Y)=Pr(j,X,Y|i), are conditional probability density functions of an information element X of the second information sequence that corresponds to a received element Y of the first information sequence after the HMM transfers from a state i to a state j, the auxiliary function being expressed as: ${{Q\left( {X_{1}^{T},X_{1,p}^{T}} \right)} = {\sum\limits_{z}{{\Psi \left( {x,X_{1,p}^{T},Y_{1}^{T}} \right)}{\log \left( {\Psi \left( {z,X_{1}^{T},Y_{1}^{T}} \right)} \right)}}}},$

where p is a number of iterations, ${{\Psi \left( {z,X_{1}^{T},Y_{1}^{T}} \right)} = {\pi_{i_{o}}{\prod\limits_{t = 1}^{T}\quad {p_{i_{t - 1}i_{t}}\left( {X_{t},Y_{1}} \right)}}}},$

 T is a number of information elements in a particular information sequence, z is a HMM state sequence i₀ ^(T), π_(i) ₀ is the probability of an initial state i₀,X₁ ^(T) is the second information sequence, X_(1,p) ^(T) is a second information sequence estimate corresponding to a pth iteration, and Y₁ ^(T) is the first information sequence.
 6. The method of claim 5, wherein the auxiliary function is expanded to be: ${Q\left( {X_{1}^{T},X_{1,p}^{T}} \right)} = {{\sum\limits_{t = 1}^{T}\quad {\sum\limits_{i = 1}^{n}\quad {\sum\limits_{j = 1}^{n}\quad {{\gamma_{t,{ij}}\left( X_{1,p}^{T} \right)}{\log \left( {p_{ij}\left( {X_{t},Y_{t}} \right)} \right)}}}}} + C}$

where C does not depend on X₁ ^(T) and γ_(t,ij)(X _(1,p) ^(T))=α_(i)(X _(1,p) ^(t−1) ,Y ₁ ^(t−1))P _(ij)(X _(t,p) ,Y _(t))β_(j)(X _(t+1,p) ^(T) ,Y _(t+1) ^(T)) where α_(i)(X_(1,p) ^(t),Y₁ ^(T)) and β_(j)(X_(t+1,p) ^(T),Y_(t+1) ^(T)) are the elements of forward and backward probability vectors defined as $\begin{matrix} {{{\alpha \left( {X_{1}^{t},Y_{1}^{t}} \right)} = {\pi {\prod\limits_{i = 1}^{t}\quad {P\left( {X_{i},Y_{i}} \right)}}}},{{and}\quad \beta \quad \left( {X_{1}^{T},Y_{1}^{T}} \right)}} \\ {{= {\prod\limits_{j = t}^{T}\quad {{P\left( {X_{j},Y_{j}} \right)}1}}},} \end{matrix}$

π is an initial probability vector, 1 is the column vector of ones.
 7. The method of claim 6, wherein a source of an encoded sequence is a trellis code modulator (TCM), the TCM receiving a source information sequence I₁ ^(T) and outputting X₁ ^(T) as an encoded information sequence that is transmitted, the TCM defining X_(t)=g_(t)(S_(t),I_(t)) where X_(t) and I_(t) are the elements of X₁ ^(T) and I₁ ^(T) for each time t, respectively, S_(t) is a state of the TCM at t, and g_(t)(·) is a function relating X_(t), to I_(t) and S_(t), the method comprising: generating, for iteration p+1, a source information sequence estimate I_(1,p+1) ^(T) that corresponds to a sequence of TCM state transitions that has a longest cumulative distance L(S_(t−1)) at t=1 or L(S₀), wherein a distance for each of the TCM state transitions is defined by L(S_(t))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))) for the TCM state transitions at each t for t=1, . . . , T and the cumulative distance is the sum of m(Î_(t)(S_(t))) for all t, m(Î_(t)(S_(t))) being defined as ${{m\left( {{\hat{I}}_{t}\left( S_{t} \right)} \right)} = {\sum\limits_{i = 1}^{n_{c}}\quad {\sum\limits_{j = 1}^{n_{c}}\quad {{\gamma_{t,{ij}}\left( I_{1,p}^{T} \right)}\log \quad {p_{c,{ij}}\left( Y_{t} \middle| {X_{t}\left( S_{t} \right)} \right)}}}}},{{{for}\quad {each}\quad t} = 1},2,\ldots \quad,T,{where}$

 X_(t)(S_(t))=g_(t)(S_(t),Î_(t)(S_(t))), n_(c) is a number of states in an HMM of the channel and p_(c,ij)(Y_(t)|X_(t)(S_(t))) are channel conditional probability density functions of Y_(t) when X_(t)(S_(t)) is transmitted by the TCM, I_(1,p+1) ^(T) being set to a sequence of Î_(t) for all t.
 8. The method of claim 7, wherein for each t=1,2, . . . , T, the method comprises: generating m(Î_(t)(S_(t))) for each possible state transition of the TCM; selecting state trajectories that correspond to largest L(S_(t))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))) for each state as survivor state trajectories; and selecting Î_(t)(S_(t))s that correspond to the selected state trajectories as I_(t,p+1)(S_(t)).
 9. The method of claim 8, further comprising: a. assigning L(S_(T))=0 for all states at t=T; b. generating m(Î_(t)(S_(t))) for all state transitions between states S_(t) and all possible states S_(t+1); c. selecting state transitions between the states S_(t) and S_(t+1) that have a largest L(S_(t))=L(S_(t−1))+m(Î_(t+1)(S_(t+1))) and Î_(t+1)(S_(t+1)) that correspond to the selected state transitions; d. updating the survivor state trajectories at states S_(t) by adding the selected state transitions to the corresponding survivor state trajectories at state S_(t+1); e. decrementing t by 1; f. repeating b-e until t=0; and g. selecting all the Î_(t)(S_(t)) that correspond to a survivor state trajectory that corresponding to a largest L(S_(t)) at t=0 as I_(1,p+1) ^(T).
 10. The method of claim 6, wherein the channel is modeled as P_(c)(Y|X)=P_(c)B_(c)(Y|X) where P_(c) is a channel state transition probability matrix and B_(c)(Y|X) is a diagonal matrix of state output probabilities, the method comprising for each t=1,2, . . . , T: generating γ_(t,i)(I_(1,p) ^(T))=α_(i)(Y₁ ^(t)|I_(1,p) ^(t))β_(i)(Y_(t+1) ^(T)|I_(t+1,p) ^(T)); selecting an Î_(t)(S_(t)) that maximizes L(S_(t))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))), where m(Î_(t)(S_(t))) is defined as ${{m\left( {{\hat{I}}_{t}\left( S_{t} \right)} \right)} = {\sum\limits_{i = 1}^{n_{c}}{{\gamma_{t,{ij}}\left( I_{1,p}^{T} \right)}{\beta_{j}\left( Y_{t} \middle| {X_{t}\left( S_{t} \right)} \right)}}}},$

n_(c) being a number of states in an HMM of the channel; selecting state transitions between states S_(t) and S_(t−1)that corresponds to a largest L(S_(t))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))); and forming survivor state trajectories by connecting selected state transitions.
 11. The method of claim 10, further comprising: selecting Î_(t)(S_(t)) that corresponds to a survivor state trajectory at t=0 that has the largest L(S_(t)) as I_(t,p+1) ^(T) for each pth iteration; comparing I_(1,p) ^(T) and I_(1,p+1) ^(T); and outputting I_(1,p+1) ^(T) as the second decode result if I_(1,p) ^(T) and I_(1,p+1) ^(T) are within the compare threshold.
 12. A maximum a posteriori (MAP) decoder that decodes a transmitted information sequence using a received information sequence received through a channel, comprising: a memory; and a controller coupled to the memory, the controller iteratively-generating decode results, X_(i), for i=1,2, . . . n, where n is an integer, and where each X_(i) is generated by employing X_(i−1), and X₁ is generated from said first information sequence and from an initial decode result, X₀, and ceasing said step of iteratively generating, and outputting last-generated decode results when difference between said last-generated decode results and next-to-last-generated decode results is within a compare threshold.
 13. A maximum a posteriori (MAP) decoder that decodes a transmitted information sequence using a received information sequence received through a channel, comprising: a memory; and a controller coupled to the memory, the controller iteratively generating a sequence of one or more decode results starting with an initial decode result, and outputting one of adjacent decode results as a decode of the input information sequence if the adjacent decode results are within a compare threshold wherein the controller: a. generates the initial decode result as a first decode result; b. generates a second decode result based on the first decode result and a model of the channel; c. compares the first and second decode results; d. replaces the first decode result with the second decode result; and e. repeats b-d until the first and second decode result are not within the compare threshold.
 14. The decoder of claim 13, wherein the controller searches for information sequence that maximizes a value of an auxiliary function.
 15. The decoder of claim 14, wherein the auxiliary function is based on expectation maximization (EM).
 16. The decoder of claim 15, wherein the model of the channel is a Hidden Markov Model (HMM) having an initial state probability vector π and probability density matrix (PDM) of P(X,Y), where XεX, YεY and elements of P(X,Y), p_(ij)(X,Y)=Pr(j,X,Y|i), are conditional probability density functions of an information element X of the second information sequence that corresponds to a received element Y of the first information sequence after the HMM transfers from a state i to a state j, the auxiliary function being expressed as: ${{Q\left( {X_{1}^{T},X_{1,p}^{T}} \right)} = {\sum\limits_{z}{{\Psi \left( {z,X_{1,p}^{T},Y_{1}^{T}} \right)}{\log \left( {\Psi \left( {z,X_{1}^{T},Y_{1}^{T}} \right)} \right)}}}},$

where p is a number of iterations, ${{\Psi \left( {z,X_{1}^{T},Y_{1}^{T}} \right)} = {\pi_{i_{o}}{\prod\limits_{t = 1}^{T}\quad {p_{i_{t - 1}i_{t}}\left( {X_{t},Y_{t}} \right)}}}},$

 T is a number of information elements in a particular information sequence, z is a HMM state sequence i₀ ^(T), π_(i) ₀ is the probability of an initial state i₀,X₁ ^(T) is the second information sequence, X_(1,p) ^(T) is a second information sequence estimate corresponding to a pth iteration, and Y₁ ^(T) is the first information sequence.
 17. The decoder of claim 16, wherein the auxiliary function is expanded to be: ${Q\left( {X_{1}^{T},X_{1,p}^{T}} \right)} = {{\sum\limits_{t = 1}^{T}\quad {\sum\limits_{i = 1}^{n}\quad {\sum\limits_{j = 1}^{n}\quad {{\gamma_{t,{ij}}\left( X_{1,p}^{T} \right)}{\log \left( {p_{ij}\left( {X_{t},Y_{t}} \right)} \right)}}}}} + C}$

where C does not depend on X₁ ^(T) and γ_(t,ij)(X _(1,p) ^(T))=α_(i)(X _(1,p) ^(t−1) ,Y ₁ ^(t−1))p _(ij)(X _(t,p) ,Y _(t))β_(j)(X _(t+1,p) ^(T) ,Y _(t+1) ^(T)) where α_(i)(X_(1,p) ^(t),Y₁ ^(T)) and β_(j)(X_(t+1,p) ^(T),Y_(t+1) ^(T)) are the elements of forward and backward if probability vectors defined as $\begin{matrix} {{{\alpha \left( {X_{1}^{t},Y_{1}^{t}} \right)} = {\pi {\prod\limits_{i = 1}^{t}\quad {P\left( {X_{i},Y_{i}} \right)}}}},{{and}\quad \beta \quad \left( {X_{t + 1}^{T},Y_{t + 1}^{T}} \right)}} \\ {{= {\prod\limits_{j = {t + 1}}^{T}\quad {{P\left( {X_{j},Y_{j}} \right)}1}}},} \end{matrix}$

π is an initial probability vector, 1 is the column vector of ones.
 18. The decoder of claim 17, wherein a source of an encoded sequence is a trellis code modulator (TCM), the TCM receiving a source information sequence I₁ ^(T) and outputting X₁ ^(T) as an encoded information sequence that is transmitted, the TCM defining X_(t)=g_(t)(S_(t),I_(t)) where X_(t) and I_(t) are the elements of X₁ ^(T) and I₁ ^(T) for each time t, respectively, S_(t) is a state of the TCM at t, and g_(t)(·) is a function relating X_(t), to I_(t) and S_(t), the controller generates, for iteration p+1, an input information sequence estimate I_(1,p+1) ^(T) that corresponds to a sequence of TCM state transitions that has a longest cumulative distance L(S_(t−1)) at t=1 or L(S₀), wherein a distance for each of the TCM state transitions is defined by L(S_(t+1))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))) for the TCM state transitions at each t for t=1, . . . , T and the cumulative distance is the sum of m(Î_(t)(S_(t))) for all t, m(Î_(t)(S_(t))) being defined as ${{m\left( {{\hat{I}}_{t}\left( S_{t} \right)} \right)} = {\sum\limits_{i = 1}^{n_{c}}\quad {\sum\limits_{j = 1}^{n_{c}}\quad {{\gamma_{t,{ij}}\left( I_{1,p}^{T} \right)}\log \quad p_{c,{ij}}\left( Y_{t} \middle| {X_{t}\left( S_{t} \right)} \right)}}}},$

for each t=1,2, . . . , T, where X_(t)(S_(t))=g_(t)(S_(t),Î_(t)(S_(t))), n_(c) is a number of states in an HMM of the channel and P_(c,ij)(Y_(t)|X_(t)(S_(t))) are channel conditional probability density functions of Y_(t) when X_(t)(S_(t)) is transmitted by the TCM, I_(1,p+1) ^(T) being set to a sequence of Î_(t) for all t.
 19. The decoder of claim 18, wherein for each t=1,2, . . . , T, the controller generating m(Î_(t)(S_(t))) for each possible state transition of the TCM, selecting state trajectories that correspond to largest L(S_(t))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))) for each state as survivor state trajectories, and selecting Î_(t+1)(S_(t+1))s that correspond to the selected state trajectories as I_(t+1,p+1)(S_(t+1)).
 20. The decoder of claim 19, wherein the controller: a. assigns L(S_(T))=0 for all states at t=T; b. generates m(Î_(t)(S_(t))) for all state transitions between states S_(t) and all possible states S_(t+1); c. selects state transitions between the states S_(t) and S_(t+1) that have a largest L(S_(t))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))) and Î_(t+1)(S_(t+1)) that correspond to the selected state transitions; d. updates the survivor state trajectories at states S_(t) by adding the selected state transitions to the corresponding survivor state trajectories at state S_(t+1); e. decrements t by 1; f. repeats b-e until t=0; and g. selects all the Î_(t)(S_(t)) that correspond to a survivor state trajectory that corresponding to a largest L(S_(t)) at t=0 as I_(1,p+1) ^(T).
 21. The decoder of claim 20, wherein the channel is modeled as P_(c)(Y|X)=P_(c)B_(c)(Y|X) where P_(c) is a channel state transition probability matrix and B_(c)(Y|X) is a diagonal matrix of state output probabilities, for each t=1,2, . . . , T, the controller: generates γ_(t,i)(I_(1,p) ^(T))=α_(i)(Y₁ ^(t)|I_(1,p) ^(t))β_(i)(Y_(t+1) ^(T)|I_(t+1,p) ^(T)); selects an Î_(t)(S_(t)) that maximizes L(S_(t))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))), where m(Î_(t)(S_(t))) is defined as ${{m\left( {{\hat{I}}_{t}\left( S_{t} \right)} \right)} = {\sum\limits_{i = 1}^{n_{c}}{{\gamma_{t,i}\left( I_{1,p}^{T} \right)}{\beta_{j}\left( Y_{t} \middle| {X_{t}\left( S_{t} \right)} \right)}}}},$

n_(c) being a number of states in an HMM of the channel; selects state transitions between states S_(t) and S_(t+1) that corresponds to a largest L(S_(t))=L(S_(t+1))+m(Î_(t+1)(S_(t+1))); and forms survivor state trajectories by connecting selected state transitions.
 22. The decoder of claim 21, wherein the controller selects Î_(t)(S_(t)) that corresponds to a survivor state trajectory at t=0 that has the largest L(S_(t)) as I_(1,p+1) ^(T) for each pth iteration, compares I_(1,p) ^(T) and I_(1,p+1) ^(T), and outputs I_(1,p+1) ^(T) as the second decode result if I_(1,p) ^(T) and I_(1,p+1) ^(T) are within the compare threshold. 